# Properties

 Label 2-1216-19.11-c1-0-12 Degree $2$ Conductor $1216$ Sign $-0.0977 - 0.995i$ Analytic cond. $9.70980$ Root an. cond. $3.11605$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (1 + 1.73i)9-s + 4·11-s + (−0.5 − 0.866i)13-s + (−0.499 − 0.866i)15-s + (−1.5 + 2.59i)17-s + (4 − 1.73i)19-s + (−2.5 − 4.33i)23-s + (2 + 3.46i)25-s − 5·27-s + (3.5 + 6.06i)29-s + 4·31-s + (−2 + 3.46i)33-s − 10·37-s + ⋯
 L(s)  = 1 + (−0.288 + 0.499i)3-s + (−0.223 + 0.387i)5-s + (0.333 + 0.577i)9-s + 1.20·11-s + (−0.138 − 0.240i)13-s + (−0.129 − 0.223i)15-s + (−0.363 + 0.630i)17-s + (0.917 − 0.397i)19-s + (−0.521 − 0.902i)23-s + (0.400 + 0.692i)25-s − 0.962·27-s + (0.649 + 1.12i)29-s + 0.718·31-s + (−0.348 + 0.603i)33-s − 1.64·37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1216$$    =    $$2^{6} \cdot 19$$ Sign: $-0.0977 - 0.995i$ Analytic conductor: $$9.70980$$ Root analytic conductor: $$3.11605$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1216} (961, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1216,\ (\ :1/2),\ -0.0977 - 0.995i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.427836286$$ $$L(\frac12)$$ $$\approx$$ $$1.427836286$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 + (-4 + 1.73i)T$$
good3 $$1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2}$$
7 $$1 + 7T^{2}$$
11 $$1 - 4T + 11T^{2}$$
13 $$1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}$$
23 $$1 + (2.5 + 4.33i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-3.5 - 6.06i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 - 4T + 31T^{2}$$
37 $$1 + 10T + 37T^{2}$$
41 $$1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-5.5 - 9.52i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (5.5 - 9.52i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (7.5 - 12.9i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−10.12974116851784177510739424791, −9.149145462644269915291853308443, −8.441077738281850344115536236223, −7.32789131671831894718956048886, −6.73413549737044777708227204023, −5.69132326049427070035321836526, −4.72791856684776711902182837348, −3.96214577637191392787099964113, −2.89103846145515756479984791114, −1.41569620587877553977564110850, 0.70474159750067721955262695646, 1.83657513146358670977229837723, 3.42643847339027405608035657755, 4.26937653309723728699824070656, 5.30244719325971240695223039780, 6.39919526618954127540283321712, 6.87775854573007508450641997933, 7.82648993991234874058193208123, 8.730656008192374881332195779733, 9.535410791987473980052116502366